Transport Equation


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9 weeks ago by
Hi there,

I have the equation
   $\frac{\partial c}{\partial t}+v_x\frac{\partial c}{\partial x}+v_y\frac{\partial c}{\partial y}-\frac{\partial}{\partial x}\left(D_{xx}\frac{\partial c}{\partial x}+D_{xy}\frac{\partial c}{\partial y}\right)-\frac{\partial}{\partial y}\left(D_{yx}\frac{\partial c}{\partial x}+D_{yy}\frac{\partial c}{\partial y}\right)=f$ct +vxcx +vycy x (Dxxcx +Dxycy )y (Dyxcx +Dyycy )=ƒ  

I understand the time stepping and the advection part. but how could I program diffusion part? especially the
 $-\frac{\partial}{\partial x}\left(D_{xy}\frac{\partial c}{\partial y}\right)$x (Dxycy ) 
here Dxx, Dyy and Dxy are known scalars.

Thank you very much for your help

 
Community: FEniCS Project

1 Answer


4
9 weeks ago by
So what you essentially have is an anisotropic diffusion term   $\nabla\cdot\left(\mathbf{D}\nabla c\right)$·(Dc) with  $\mathbf{D}$D a tensor of (constant) coefficients.
Implementing such a term in FEniCS is as easy as (after integration by parts, using a CG basis)

from dolfin import *

mesh = UnitSquareMesh(2,2)

# Tensor with (isotropic) diffusion coefficients
D = Constant([[1.,1.],[1., 1.]])

V = FunctionSpace(mesh,'CG', 1)
c = TrialFunction(V)
w = TestFunction(V)

a = inner(D*grad(c), grad(w))*dx
A = assemble(a)​
It works perfect, Thank you very much
written 9 weeks ago by Ruben Gonzalez  
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